The points $A \ne B$ are given on the plane. The point $C$ moves along the plane in such a way that $\angle ACB = \alpha$ , where $\alpha$ is the fixed angle from the interval ($0^o, 180^o$). The circle inscribed in triangle $ABC$ has center the point $I$ and touches the sides $AB, BC, CA$ at points $D, E, F$ accordingly. Rays $AI$ and $BI$ intersect the line $EF$ at points $M$ and $N$, respectively. Show that: a) the segment $MN$ has a constant length, b) all circles circumscribed around triangle $DMN$ have a common point

Let two circles be externally tangent at point $C$, with parallel diameters $A_1A_2, B_1B_2$ (i.e. the quadrilateral $A_1B_1B_2A_2$ is a trapezoid with bases $A_1A_2$ and $B_1B_2$ or parallelogram). Circle with the center on the common internal tangent to these two circles, passes through the intersection point of lines $A_1B_2$ and $A_2B_1$ as well intersects those lines at points $M, N$. Prove that the line $MN$ is perpendicular to the parallel diameters $A_1A_2, B_1B_2$. (Yuri Biletsky)

Let three circles be externally tangent in pairs, with parallel diameters $A_1A_2, B_1B_2, C_1C_2$ (i.e. each of the quadrilaterals $A_1B_1B_2A_2$ and $A_1C_1C_2A_2$ is a parallelogram or trapezoid, which segment $A_1A_2$ is the base). Prove that $A_1B_2, B_1C_2, C_1A_2$ intersect at one point. (Yuri Biletsky )

In the triangle $ABC$ with sides $BC> AC> AB$ the angles between altiude and median drawn from one vertex are considered. Find out at which vertex this angle is the largest of the three. (Rozhkova Maria)

The circles ${{w} _ {1}}$ and ${{w} _ {2}}$ intersect at points $P$ and $Q$. Let $AB$ and $CD$ be parallel diameters of circles ${ {w} _ {1}}$ and ${{w} _ {2}} $, respectively. In this case, none of the points $A, B, C, D$ coincides with either $P$ or $Q$, and the points lie on the circles in the following order: $A, B, P, Q$ on the circle ${{w} _ {1} }$ and $C, D, P, Q$ on the circle ${{w} _ {2}} $. The lines $AP$ and $BQ$ intersect at the point $X$, and the lines $CP$ and $DQ$ intersect at the point $Y, X \ne Y$. Prove that all lines $XY$ for different diameters $AB$ and $CD$ pass through the same point or are all parallel. (Serdyuk Nazar)

Given a triangle $ ABC $, $ AD $ is its angle bisector. Let $ E, F $ be the centers of the circles inscribed in the triangles $ ADC $ and $ ADB $, respectively. Denote by $ \omega $, the circle circumscribed around the triangle $ DEF $, and by $ Q $, the intersection point of $ BE $ and $ CF $, and $ H, J, K, M $ , respectively the second intersection point of the lines $ CE, CF, BE, BF $ with circle $ \omega $. Let $\omega_1, \omega_2 $ the circles be circumscribed around the triangles $ HQJ $ and $ KQM $ Prove that the intersection point of the circles $\omega_1, \omega_2 $ different from $ Q $ lies on the line $ AD $. (Kivva Bogdan)

Let $ H $ be the intersection point of the altitudes $ AP $ and $ CQ $ of the acute-angled triangle $ ABC $. On its median $ BM $ marked points $ E $ and $ F $ so that $ \angle APE = \angle BAC $ and $ \angle CQF = \angle BCA $, and the point $ E $ lies inside the triangle $ APB $, and the point $ F $ lies inside the triangle $ CQB $. Prove that the lines $ AE $, $ CF $ and $ BH $ intersect at one point. (Vyacheslav Yasinsky)

Three circles are constructed for the triangle $ABC $: the circle ${{w} _ {A}} $ passes through the vertices $B $ and $C $ and intersects the sides $AB $ and $ AC $ at points ${{A} _ {1}} $ and ${{A} _ {2}} $ respectively, the circle ${{w} _ {B}} $ passes through the vertices $A $ and $C $ and intersects the sides $BA $ and $BC $ at the points ${{B} _ {1}} $ and ${{B} _ {2}} $, ${{w} _ {C}} $ passes through the vertices $A $ and $B $ and intersects the sides $CA $ and $CB $ at the points ${{C} _ {1}} $ and ${{C} _ {2}} $. Let ${{A} _ {1}} {{A} _ {2}} \cap {{B} _ {1}} {{B} _ {2}} = {C} '$, ${{A} _ {1}} {{A} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {B} '$ ta ${ {B} _ {1}} {{B} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {A} '$ is Prove that the perpendiculars, which are omitted from the points ${A} ', \, \, {B}', \, \, {C} '$ to the lines $BC $, $CA $ and $AB $ respectively intersect at one point. (Rudenko Alexander)

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers ${{O} _ {1}}$ and ${{O} _ {2}}$ intersect at points $A$ and $B$, respectively. The line ${{O} _ {1}} {{O} _ {2}}$ intersects ${{w} _ {1}}$ at the point $Q$, which does not lie inside the circle ${{w} _ {2}}$, and ${{w} _ {2}}$ at the point $X$ lying inside the circle ${{w} _ {1} }$. Around the triangle ${{O} _ {1}} AX$ circumscribe a circle ${{w} _ {3}}$ intersecting the circle ${{w} _ {1}}$ for the second time in point $T$. The line $QT$ intersects the circle ${{w} _ {3}}$ at the point $K$, and the line $QB$ intersects ${{w} _ {2}}$ the second time at the point $H$. Prove that a) points $T, \, \, X, \, \, B$ lie on one line; b) points $K, \, \, X, \, \, H$ lie on one line. (Vadym Mitrofanov)

The line passing through the center of the equilateral triangle $ ABC $ intersects the lines $ AB $, $ BC $ and $ CA $ at the points $ {{C} _ {1}} $, $ {{A} _ {1}} $ and $ {{B} _ {1}} $, respectively. Let $ {{A} _ {2}} $ be a point that is symmetric $ {{A} _ {1}} $ with respect to the midpoint of $ BC $; the points $ {{B} _ {2}} $ and $ {{C} _ {2}} $ are defined similarly. Prove that the points $ {{A} _ {2}} $, $ {{B} _ {2}} $ and $ {{C} _ {2}} $ lie on the same line tangent to the inscribed circle of the triangle $ ABC $. (Serdyuk Nazar)

On the horizontal line from left to right are the points $P, \, \, Q, \, \, R, \, \, S$. Construct a square $ABCD$, for which on the line $AD$ lies lies the point $P$, on the line $BC$ lies the point $Q$, on the line $AB$ lies the point $R$, on the line $CD$ lies the point $S $.

Circles $w_1$ and $w_2$ with centers at points $O_1$ and $O_2$ respectively, intersect at points $A$ and $B$. A line passing through point $B$, intersects the circles $w_1$ and $w_2$ at points $C$ and $D$ other than $B$. Tangents to the circles $w_1$ and $w_2$ at points $C$ and $D$ intersect at point $E$. Line $EA$ intersects the circumscribed circle $w$ of triangle $AO_1O_2$ at point $F$. Prove that the length of the segment is $EF$ is equal to the diameter of the circle $w$. (Vovchenko V., Plotnikov M.)

The median $CM$ is drawn in the triangle $ABC$ intersecting bisector angle $BL$ at point $O$. Ray $AO$ intersects side $BC$ at point $K$, beyond point $K$ draw the segment $KT = KC$. On the ray $BC$ beyond point $C$ draw a segment $CN = BK$. Prove that is a quadrilateral $ABTN$ is cyclic if and only if $AB = AK$. (Vladislav Yurashev)

In the acute triangle $ABC$ the orthocenter $H$ and the center of the circumscribed circle $O$ were noted. The line $AO$ intersects the side $BC$ at the point $D$. A perpendicular drawn to the side $BC$ at the point $D$ intersects the heights from the vertices $B$ and $C$ of the triangle $ABC$ at the points $X$ and $Y$ respectively. Prove that the center of the circumscribed circle $\Delta HXY$ is equidistant from the points $B$ and $C$. (Danilo Hilko)

The point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AC$ intersects the circumscribed circle $\Delta ABO$ for second time at the point $X$. Prove that $XO \bot BC$.

In the quadrilateral $ABCD $, $AB = BC $, the point $K $ is the midpoint of the side $CD $, the rays $BK $ and $AD $ intersect at the point $M $ , the circumscribed circle $ \Delta ABM $ intersects the line $AC $ for the second time at the point $P $. Prove that $\angle BKP = 90 {} ^ \circ $. (Anton Trygub)

Denote in the triangle $ABC$ by $T_A,T_B,T_C$ the touch points of the exscribed circles of $\vartriangle ABC$, tangent to sides $BC, AC$ and $AB$ respectively. Let $O$ be the center of the circumcircle of $\vartriangle ABC$, and $I$ is the center of it's inscribed circle. It is known that $OI\parallel AC$. Prove that $\angle T_A T_B T_C= 90^o - \frac12 \angle ABC$. (Anton Trygub)

Let $ABCDE$ be a regular pentagon with center $M$. Point $P \ne M$ is selected on segment $MD$. The circumscribed circle of triangle $ABP$ intersects the line $AE$ for second time at point $Q$, and a line that is perpendicular to the $CD$ and passes through $P$, for second time at the point $R$. Prove that $AR = QR$.

The line $\ell$ is perpendicular to the side $AC$ of the acute triangle $ABC$ and intersects this side at point $K$, and the circumcribed circle $\vartriangle ABC$ at points $P$ and $T$ (point P on the other side of line $AC$, as the vertex $B$). Denote by $P_1$ and $T_1$ - the projections of the points $P$ and $T$ on line $AB$, with the vertices $A, B$ belong to the segment $P_1T_1$. Prove that the center of the circumscribed circle of the $\vartriangle P_1KT_1$ lies on a line containing the midline $\vartriangle ABC$, which is parallel to the side $AC$. (Anton Trygub)

It is known that in the triangle $ABC$ the smallest side is $BC$. Let $X, Y, K$ and $L$ - points on the sides $AB, AC$ and on the rays $CB, BC$, respectively, are such that $BX = BK = BC =CY =CL$. The line $KX$ intersects the line $LY$ at the point $M$. Prove that the intersection point of the medians $\vartriangle KLM$ coincides with the center of the inscribed circle $\vartriangle ABC$.

Let $M$ be the midpoint of the side $AC$ of triangle $ABC$. Inside $\vartriangle BMC$ was found a point $P$ such that $\angle BMP = 90^o$, $\angle ABC+ \angle APC =180^o$. Prove that $\angle PBM + \angle CBM = \angle PCA$. (Anton Trygub)

A point $P$ was chosen on the smaller arc $BC$ of the circumcircle of the acute-angled triangle $ABC$. Points $R$ and $S$ on the sides$ AB$ and $AC$ are respectively selected so that $CPRS$ is a parallelogram. Point $T$ on the arc $AC$ of the circumscribed circle of $\vartriangle ABC$ such that $BT \parallel CP$. Prove that $\angle TSC = \angle BAC$. (Anton Trygub)

Inside the quadrilateral $ABCD$ marked a point $O$ such that $\angle OAD+ \angle OBC = \angle ODA + \angle OCB = 90^o$. Prove that the centers of the circumscribed circles around triangles $OAD$ and $OBC$ as well as the midpoints of the sides $AB$ and $CD$ lie on one circle. (Anton Trygub)

Let $ABCD$ be an isosceles trapezoid, $AD=BC$, $AB \parallel CD$. The diagonals of the trapezoid intersect at the point $O$, and the point $M$ is the midpoint of the side $AD$. The circle circumscribed around the triangle $BCM$ intersects the side $AD$ at the point $K$. Prove that $OK \parallel AB$.

In the triangle $ABC$, the altitude $BH$ and the angle bisector $BL$ are drawn, the inscribed circle $w$ touches the side of the $AC$ at the point $K$. It is known that $\angle BKA = 45^o$. Prove that the circle with diameter $HL$ touches the circle $w$. (Anton Trygub)

Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ have no common points. The line$ AB$ is a common internal tangent, and the line $CD$ is a common external tangent to these circles, where $A, C \in k_1$ and $B, D \in k_2$. Knowing that $AB=12$ and $CD =16$, find the value of the product $r_1r_2$.

Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Prove that if $\frac{H_BC}{AC} = \frac{H_CA}{AB}$, then the line symmetric to $BC$ with respect to line $H_BH_C$ is tangent to the circumscribed circle of triangle $H_BH_CA$. (Proposed by Mykhailo Bondarenko)

Let $ABCD$ be the cyclic quadrilateral. Suppose that there exists some line $l$ parallel to $BD$ which is tangent to the inscribed circles of triangles $ABC, CDA$. Show that $l$ passes through the incenter of $BCD$ or through the incenter of $DAB$. (Proposed by Fedir Yudin)